The function \(f(x) =\log_e (\sin x), x \in (0,\pi)\) is
(A) strictly increasing on \(\left(0,\displaystyle\frac{\pi}{2}\right)\)
(B) strictly decreasing on \(\left(0,\displaystyle\frac{\pi}{2}\right)\)
(C) strictly increasing on \(\left(\displaystyle\frac{\pi}{2},\pi\right)\)
(D) strictly decreasing on \(\left(\displaystyle\frac{\pi}{2},\pi\right)\)
(E) strictly increasing on \((0,\pi)\)
Choose the correct answer from the options given below.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (D) only
By the chain rule:
\( \displaystyle f'(x) = \frac{1}{\sin x} \cdot \cos x = \cot x \)
So \(f'(x) = \cot x \gt 0\) on \(\left(0, \frac{\pi}{2}\right)\) and \(f'(x) = \cot x \lt 0\) on \(\left(\frac{\pi}{2}, \pi\right)\).
Checking the statements
(A) strictly increasing on \(\left(0,\frac{\pi}{2}\right)\) — Correct. \(f'(x) = \cot x \gt 0\) there.
(B) strictly decreasing on \(\left(0,\frac{\pi}{2}\right)\) — Incorrect. \(f'(x) = \cot x \gt 0\) there, so f increases.
(C) strictly increasing on \(\left(\frac{\pi}{2},\pi\right)\) — Incorrect. \(f'(x) = \cot x \lt 0\) there.
(D) strictly decreasing on \(\left(\frac{\pi}{2},\pi\right)\) — Correct. \(f'(x) = \cot x \lt 0\) there.
(E) strictly increasing on \((0,\pi)\) — Incorrect. f decreases on \(\left(\frac{\pi}{2},\pi\right)\).
NCERT (2026-27 edition):
Increasing and Decreasing Functions
